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Theorem equvini 1927
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
equvini  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )

Proof of Theorem equvini
StepHypRef Expression
1 equcomi 1646 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
21alimi 1546 . . . . 5  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
3 a9e 1891 . . . . 5  |-  E. z 
z  =  y
42, 3jctir 524 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  x  =  z  /\  E. z  z  =  y
) )
54a1d 22 . . 3  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( A. z  x  =  z  /\  E. z  z  =  y ) ) )
6 19.29 1583 . . 3  |-  ( ( A. z  x  =  z  /\  E. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
75, 6syl6 29 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
8 a9e 1891 . . . . . 6  |-  E. z 
z  =  x
91eximi 1563 . . . . . 6  |-  ( E. z  z  =  x  ->  E. z  x  =  z )
108, 9ax-mp 8 . . . . 5  |-  E. z  x  =  z
1110a1ii 24 . . . 4  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z  x  =  z ) )
1211anc2ri 541 . . 3  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y ) ) )
13 19.29r 1584 . . 3  |-  ( ( E. z  x  =  z  /\  A. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
1412, 13syl6 29 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
15 ioran 476 . . 3  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  <->  ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y ) )
16 nfeqf 1898 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
17 ax-8 1643 . . . . . 6  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1817anc2li 540 . . . . 5  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1918equcoms 1651 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
2016, 19spimed 1917 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
2115, 20sylbi 187 . 2  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
227, 14, 21ecase3 907 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  equvin  1941  sbequi  1999  a12lem2  29131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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